A t-spanner of a graph G is a subgraph H of G, which the distance between any two vertices in H is at most t times their distance in G. A tree t-spanner of G is a t-spanner which is a tree. In this dissertation, we discuss the t-spanners on trapezoid, permutation, and interval graphs. We first introduce an O(n) algorithm for finding a tree 4-spanner on trapezoid graphs. Then, give an O(n)algorithm for finding a tree 3-spanner on permutation graphs, improving the existed O(n + m) algorithm. Since the class of permutation graphs is a subclass of trapezoid graphs, we can apply the algorithm on permutation graphs to find the approximation of a tree 3-spanner on trapezoid graphs in O(n) time with edge bound 2n. Finally, we show that not all interval graphs have a tree 2-spanner.